Theorem spw 2032

Description: Weak version of the specialization scheme sp 2182. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2182 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2182 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2134 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2182 are spfw 2031 (minimal distinct variable requirements), spnfw 1978 (when 𝑥 is not free in ¬ 𝜑), spvw 1979 (when 𝑥 does not appear in 𝜑), sptruw 1805 (when 𝜑 is true), spfalw 1996 (when 𝜑 is false), and spvv 1995 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1909	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1909	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1909	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2031	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by:  hba1w  2046  19.8aw  2049  exexw  2050  spaev  2051  ax12w  2132  rspw  3235  reldisj  4452  ralidmw  4507  dtruALT2  5369  dtruOLD  5445  bj-ssblem1  36656  bj-ax12w  36679  eu6w  42691